Free cooling and high-energy tails of granular gases with variable restitution coefficient

We prove the so-called generalized Haff's law yielding the optimal algebraic cooling rate of the temperature of a granular gas described by the homogeneous Boltzmann equation for inelastic interactions with non constant restitution coefficient. Our analysis is carried through a careful study of the infinite system of moments of the solution to the Boltzmann equation for granular gases and precise Lp estimates in the selfsimilar variables. In the process, we generalize several results on the Boltzmann collision operator obtained recently for homogeneous granular gases with constant restitution coefficient to a broader class of physical restitution coefficients that depend on the collision impact velocity. This generalization leads to the so-called L1-exponential tails theorem. for this model

Propagation of L1L^{1} and L∞L^{\infty} Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann Equation

We consider the $n$-dimensional space homogeneous Boltzmann equation for elastic collisions for variable hard potentials with Grad (angular) cutoff. We prove sharp moment inequalities, the propagation of $L^1$-Maxwellian weighted estimates, and consequently, the propagation $L^\infty$-Maxwellian weighted estimates to all derivatives of the initial value problem associated to the afore mentioned problem. More specifically, we extend to all derivatives of the initial value problem associated to this class of Boltzmann equations corresponding sharp moment (Povzner) inequalities and time propagation of $L^1$-Maxwellian weighted estimates as originally developed A.V. Bobylev in the case of hard spheres in 3 dimensions; an improved sharp moments inequalities to a larger class of angular cross sections and $L^1$-exponential bounds in the case of stationary states to Boltzmann equations for inelastic interaction problems with `heating' sources, by A.V. Bobylev, I.M. Gamba and V.Panferov, where high energy tail decay rates depend on the inelasticity coefficient and the the type of `heating' source; and more recently, extended to variable hard potentials with angular cutoff by I.M. Gamba, V. Panferov and C. Villani in the elastic case collision case and so $L^1$-Maxwellian weighted estimated were shown to propagate if initial states have such property. In addition, we also extend to all derivatives the propagation of $L^\infty$-Maxwellian weighted estimates to solutions of the initial value problem to the Boltzmann equations for elastic collisions for variable hard potentials with Grad (angular) cutoff.Comment: 24 page

Klein-Gordon equation from Maxwell-Lorentz dynamics

We consider Maxwell-Lorentz dynamics: that is to say, Newton's law under the action of a Lorentz's force which obeys the Maxwell equations. A natural class of solutions are those given by the Lagrangian submanifolds of the phase space when it is endowed with the symplectic structure modified by the electromagnetic field. We have found that the existence of this type of solution leads us directly to the Klein-Gordon equation as a compatibility condition. Therefore, surprisingly, quite natural assumptions on the classical theory involve a quantum condition without any process of limit. This result could be a partial response to the inquiries of Dirac.Comment: 8 pages; some misprints corrected; added some coments in the abstract and also explaining the relationship with the work of P.A.M. Dirac; also added 2 reference